Jacobi Elliptic Functions
from a Dynamical Systems
Point of View
Original Paper by Kenneth R. Meyer
Project and Presentation by Andrew Featherston and
Jesse Kreger
Topic Overview and Results
•
•
•
•
We will examine three Jacobi Elliptic Functions that can
be defined by a three dimensional system of ordinary
differential equations.
We will analyze this three dimensional system using
techniques we have learned in this class.
We will show how we can use Jacobi Elliptic Functions
to compute a difficult integral.
Thesis: Jacobi Elliptic Functions can be defined as a set of functions that satisfy a three
dimensional first order system of differential equations. These functions can then be used in a
wide variety of scientific contexts, such as dealing with certain algebraic expressions.
What are Jacobi Elliptic Functions?
Properties of Jacobi Elliptic Functions
•
•
•
Group of basic elliptic functions
Similar to trigonometric functions
Used to anti-differentiate certain expressions
and have many uses in real world sciences
such as physics, chemistry, and engineering.
Jacobi Elliptic Functions
•
There are three Jacobi Elliptic Functions we will be
dealing with:
o Sine amplitude = sn(t,k)
o Cosine amplitude = cn(t,k)
o Delta amplitude = dn(t,k)
Functions Defined By ODEs
We can use a differential equation approach to define functions. We will use
the trigonometric functions cosine and sine as one such example.
Our approach is to define cosine and sine as
the harmonic oscillating functions that satisfy
this two dimensional system.
Jacobi Elliptic Functions as a System of ODEs
By letting x=sn(t,k) y=cn(t,k)
z=dn(t,k) we have:
And since our initial conditions are
satisfied as well, our system is
satisfied.
Proposition:
•
As k approaches 0 from the right we have:
o
o
o
sn(t,k)
cn(t,k)
dn(t,k)
sin(t)
cos(t)
1
Proof
When k=O our third differential becomes O.
Therefore, z=dn(t,k) is a constant and we are
left with the following two dimensional
system assuming z=1:
Proof Continued
This is a simple coupled system of ordinary linear differential equations.
Using techniques learned in this class we can find the general solution.
Proof Continued
Using Euler’s formula and Section 3.4 of our textbook we
can find that the solution to our system will be:
And thus we have shown x=sin(t) and y=cos(t).
Application: Integrating Factorable Quartics
Real World Example
•
•
•
•
•
•
Jesse decided to build an arch
Area between top of arch and the xaxis modeled by F(x)
Andrew Featherston is a blimp
aficionado, wants to see arch in
blimp
For safety reasons Jesse worried
about average distance between
blimp and arch
x-axis from 0 to 1 is path of blimp
Average distance of blimp from arch
will be F(x)/1=F(x)
Conclusion
•
•
Jacobi Elliptic Functions can be defined in terms of a
system of differential equations
o Adds depth and theory to elliptic functions
o Use basic ODEs theorems, ideas, and techniques
Jacobi Elliptic Functions have many applications to
math and sciences
o Anti-differentiation
o Physics
o Engineering
Works Cited
Blanchard, Paul, Robert L. Devaney, and Glen R. Hall. Differential Equations.
Boston, MA: Brooks/Cole, Cengage Learning, 2012. Print.
Meyer, Kenneth R. "Jacobi Elliptic Functions from a Dynamical Systems
Point of View." The American Mathematical Monthly 108.8 (2001): 729-37.
JSTOR. Web. 10 Oct. 2013.
National Institute of Standards and Technology. Jacobian Elliptic Functions
Properties. Digital Library of Mathematical Functions. Web. 6 Nov. 2013.
Weisstein, Eric W. "Jacobi Elliptic Functions." Wolfram MathWorld. Web. 6
Nov. 2013.